max, min, inflections

. Let f(x) = x2ex
(a)Find all of the local maxima and minima
(b)find point of inflection
(c)determine largest interval in which f(x) is concave downward.
(d) evaluate lin x->-inf f(x) and lim x->inf f(X)

If a function is continous everywhere then the relative extrema occur either when it is not differenciable or the derivative is zero. The function is differenciable everywhere.
We search where it is zero.
Product rule,
Now, thus,
Since, divide by it,
Thus,
Thus,.

That means the function is monotone on,
Check the signs of these intervals,
By the first derivative test is relative maximum.
By the first derivative test is relative minimum.